From this post the dial represents:
How long time has passed since you last rested.
If it's 0 then that means you spent less than 24 hours game time in the world from your last rest.
I also believe that the two yellow colors indicate the sunrise and sunset (the bottom yellow/orange color is the sunset, the dark blue afterwards is night, the top brighter yellow is the sun rise, the light blue after it is day).
For combat differences:
According to the Wikia, Easy Mode will make enemies weaker and have Carbuncle prevent you from dying numerous times before you actually die.
Final Fantasy XV also has a standard difficulty and easy mode. The player can change difficulty at any time. The easy mode is made so that one can generally win by just button mashing. In the easy mode, Carbuncle can appear to heal Noctis if he is in danger of dying.
Note that you can still die on Easy mode. See this thread:
Many will tell you it's impossible to die in easy mode but that simply is not true. It's tougher due to Carbuncle reviving you a ton of times, but i've tested this, and you can still die in easy mode because he will stop reviving you eventually.
I went up against an Iron Giant on easy and still died after only being revived a good 3-5 times. After that, Carbuncle just stopped showing up which led to a game over.
Lastly, it appears that you can extend the amount of time you can spend in Wait Mode if you are playing on the Easy difficulty. See here:
If you want to have a longer wait mode, you can always toggle the difficulty settings. To do this, enter the pause menu, select Difficulty, then switch to easy. With a lower difficulty setting, enemies are weaker and the wait mode is extended, giving you a longer time to rehash your decisions.
As for trophies, there is a single missable trophy if you play on Easy Mode. From this list:
Chosen King: Defeated Ifrit on Normal difficulty.
Though, I've read that you can switch the difficulty to Normal just before the fight and still get the trophy.
Finally, for your question about EXP or items rewarded, I can't find any evidence of the drop rates/amounts being any different across difficulties. From the same thread as earlier:
As far as I know, yes. But I wasn't really counting the EXP when doing my Carbuncle tests on easy.
Best Answer
George Willcox already provided the correct source to this, but didn't answer the question.
The summons become available randomly.
As explained in the Wiki:Astral, certain circumstances increase the chance for certain summons.
Titan's chance increases the more allies are KO or in critical health. Also he needs an open area to be able to appear.
Ramuh's chance increases with battle duration and does not have additional requirements.
Leviathan's chance increases when Noctis is in critical health. She needs an open area as well, but also water nearby.
Shiva's chance increases the more allies are KO or in critical health and with battle duration. She also needs space, but it is less restrictive than for Titan and Leviathan.
Bahamut is a special case and can only be summoned at a particular point in the story.